THE DESIGN AND ANALYSIS OF EXPERIMENTS. By Oscar Kempthorne. New York, John Wiley and Sons, Inc., 1952. 631 pp. $8.50. This book by a teacher of statistics (as well as a consultant for \"experimenters\") is a comprehensive study of the philosophical background for the statistical design of experiment. It is necessary to have some facility with algebraic notation and manipulation to be able to use the volume intelligently. The problems are presented from the theoretical point of view, without such practical examples as would be helpful for those not acquainted with mathematics. The mathematical justification for the techniques is given. As a somewhat advanced treatment of the design and analysis of experiments, this volume will be interesting and helpful for many who approach statistics theoretically as well as practically. With emphasis on the \"why,\" and with description given broadly, the author relates the subject matter to the general theory of statistics and to the general problem of experimental inference. MARGARET J. ROBERTSON

The Design and Analysis of Experiments

Statistics for Spatial Data.

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Convex Analysisの二,三の進展について

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

When we use simulation to estimate the performance of a stochastic system, lack of fidelity in the random input models can lead to poor system performance estimates. Since the components of many complex systems could be dependent, we want to build input models that faithfully capture such key properties. In this paper, we use the flexible NORmal To Anything (NORTA) representation for dependent inputs. However, to use the NORTA representation we need to estimate the marginal distribution parameters and a correlation matrix from real-world data, introducing input uncertainty. To quantify this uncertainty, we employ the bootstrap to capture the parameter estimation error and an equation-based stochastic kriging metamodel to propagate the input uncertainty to the output mean. Asymptotic analysis provides theoretical support for our approach, while an empirical study demonstrates that it has good finite-sample performance.

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Statistical uncertainty analysis for stochastic simulation with dependent input models

A useful performance measure on which to compare manufacturing systems is a quantile of the cycle time distribution. Unfortunately, aside from order statistic estimates, which can require significant data storage, the distribution of quantile estimates has not been shown to be normally distributed, violating a common assumption amongst ranking-and-selection (R&S) procedures. To address this, we provide empirical evidence supporting an approach using the mean of a group of quantile estimates as the comparison measure. The approach is detailed and illustrated through experimentation on four M/M/1 queues in which the 0.9 cycle-time quantile is the performance measure. Results in terms of simulation effort and accuracy are reported and compared to results obtained using the macro-replications approach for inducing normality as well as to results obtained by applying R&S procedures to quantile estimates directly. The suggested procedure is shown to provide significant savings in simulation effort while sacrificing very little in accuracy.

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Using quantiles in ranking and selection procedures

We develop and evaluate a two-level simulation procedure that produces a confidence interval for tail conditional expectation, otherwise known as conditional tail expectation. This risk measure is closely related to conditional value-at-risk, expected shortfall, and worst conditional expectation. The outer level of simulation generates risk factors and the inner level estimates each expected loss conditional on the risk factor. Our procedure uses the statistical theory of empirical likelihood to construct a confidence interval, and it uses tools from the ranking-and-selection literature to make the simulation efficient.

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A confidence interval for tail conditional expectation via two-level simulation

A simulation-based methodology is proposed to map the mean of steady-state cycle time (CT) as a function of throughput (TH) and product mix (PM) for manufacturing systems. Nonlinear regression models motivated by queueing analysis are assumed for the underlying response surface. To ensure efficiency and control estimation error, simulation experiments are built up sequentially using a multi-stage procedure to collect data for fitting the models. The resulting response surface is able to provide a CT estimate for any TH and any PM, and thus allows the decision maker to instantly investigate options and trade offs regarding their production planning.

Metamodelling for cycle time-throughput-product mix surfaces using progressive model fitting

Risk measurement involves estimating some functional of a loss distribution. This calls for nested simulation, in which risk factors are sampled at an outer level of simulation, while the inner level of simulation provides estimates of loss given each realization of the risk factors. We present a general method for providing a confidence interval for the risk measurement. It involves combining a confidence region for the losses with the confidence interval that would be used if the losses were known. This method could be very time-consuming, so we discuss ways of improving its efficiency. One is to choose the structure of the confidence region for the losses in a way that is tailored to the particular risk measure. Another involves varying the simulation effort expended on estimating the losses associated with different risk factors, even screening out many of the risk factors after slight simulation effort has revealed them to be unlikely to have much impact on the estimated risk measure. We will discuss two-level simulation in general and a specific procedure for estimating a confidence interval for tail conditional expectation.

Barry L. Nelson

Papers

Statistical uncertainty analysis for stochastic simulation with dependent input models

Using quantiles in ranking and selection procedures

A confidence interval for tail conditional expectation via two-level simulation

Metamodelling for cycle time-throughput-product mix surfaces using progressive model fitting

Two-Level Simulations for Risk Management